How Do You Calculate Elastic Potential Energy in a Spring?

For both SL and HL, calculate the elastic potential energy stored in a spring using $E_{\text{H}}=\frac{1}{2}k(\Delta x)^2$, where $k$ is the spring constant and $\Delta x$ is the extension or compression measured from equilibrium. The result is measured in joules.

Why This Equation Works

Elastic potential energy is stored when a spring is deformed. According to Hooke's law, the magnitude of its restoring force is

$$F=k\Delta x.$$

The force is not constant during deformation: it increases linearly from zero to $k\Delta x$. The stored energy equals the work done on the spring, which is the area under a force-extension graph.

For a spring obeying Hooke's law, the graph is triangular, so

$$E_{\text{H}}=\frac{1}{2}(\Delta x)(k\Delta x)=\frac{1}{2}k(\Delta x)^2.$$

An equivalent check uses the average force. Because the force rises uniformly from $0$ to $k\Delta x$, its average value is $\frac{1}{2}k\Delta x$. Multiplying this by the displacement gives the same expression. This also explains why using the final force multiplied by extension would overestimate the energy by a factor of two.

Use SI units: $k$ in $\text{N m}^{-1}$, $\Delta x$ in $\text{m}$, and energy in joules ($\text{J}$). The equation applies within the Hooke's-law region; beyond the limit of proportionality, determine the energy from the graph's area.

Worked Example

A spring has $k=250\ \text{N m}^{-1}$ and is extended by $8.0\ \text{cm}$. First convert the extension:

$$\Delta x=8.0\ \text{cm}=0.080\ \text{m}.$$

Then substitute:

$$E_{\text{H}}=\frac{1}{2}(250)(0.080)^2=0.80\ \text{J}.$$

Therefore, $0.80\ \text{J}$ of energy is stored, assuming negligible energy dissipation. The same equation applies to compression because the displacement is squared.

Common Misconceptions

Do not use the spring's total length: use its change in length from equilibrium. Do not forget to convert centimetres into metres or square the extension. Since $E_{\text{H}}\propto(\Delta x)^2$, doubling the deformation produces four times the stored energy.

IB Exam Technique

This is Topic A.3, Work, energy and power. For a Calculate question, show the equation, SI conversion, substitution, and answer. If given a force-extension graph, find the area under it instead of automatically assuming a linear relationship.